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Implicit Bilevel Optimization: Differentiating through Bilevel Optimization Programming
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Model adversarial noise and control as a bilevel interaction instead of replacing the outer problem with an OptNet best response | Optimal Control · 10 and 30 steps | average cost: 0.2722 -> 0.2379; 0.2511 -> 0.2181 | Helped |
| Train with a bilevel discrete feature layer and attacker | CIFAR-10 adversarial attack · L∞ ≤ 0, 5, 10, 15, 20, 25, 30 | accuracy: 64.0 ± 0.4 vs 63.6 ± 0.5; 44.5 ± 0.2 vs 44.3 ± 1.0; 25.3 ± 0.8 vs 24.3 ± 1.0; 15.6 ± 0.7 vs 14.6 ± 0.7; 10.0 ± 0.6 vs 9. | Helped |
| Apply a bilevel discrete layer after vanilla-network training | CIFAR-10 adversarial attack · L∞ ≤ 0, 5, 10 | accuracy: 63.6 ± 0.5 vs 63.4 ± 0.7; 44.3 ± 1.0 vs 43.8 ± 1.2; 24.2 ± 1.0 vs 24.3 ± 1.0 | Mixed |
| Increase discrete-layer attack from Q=0 to Q=10 | CIFAR-10 bilevel min-max network · Adversarial attack at the discrete layer | accuracy | No clear effect |
| Include interaction between the outer and inner interdiction problems using Bigrad(BB) rather than BB-1 | Shortest Path with Interdiction · 12×12, 18×18, and 24×24 maps, validation | accuracy: 91.7 ± 0.1 -> 94.5 ± 0.2; 94.5 ± 0.1 -> 96.4 ± 0.2; 96.3 ± 0.2 -> 97.8 ± 0.0 | Helped |
| Use Bigrad(BB) rather than the cheaper Bigrad(PT) gradient estimator | Shortest Path with Interdiction · 12×12, 18×18, and 24×24 maps, validation | accuracy: 91.6 ± 0.1 -> 94.5 ± 0.2; 94.2 ± 0.1 -> 96.4 ± 0.2; 95.6 ± 0.1 -> 97.8 ± 0.0 | Helped |
| Use the Bigrad(BB) gradient instead of the single-level BB-1 gradient | TSP with Interdiction · k=8, 10, and 12, validation | accuracy: 83.9 ± 0.4 -> 89.4 ± 0.2; 87.5 ± 0.4 -> 92.0 ± 0.1; 89.3 ± 0.1 -> 93.5 ± 0.2 | Helped |
| Replace Bigrad(BB) with the lower-computation Bigrad(PT) estimator | TSP with Interdiction · k=8, 10, and 12, validation | accuracy: 89.4 ± 0.2 vs 89.4 ± 0.1; 92.0 ± 0.1 vs 91.9 ± 0.1; 93.5 ± 0.2 vs 93.7 ± 0.1 | No clear effect |
| Use the single-level BB-1 estimator instead of the single-level PT-1 estimator | Shortest Path with Interdiction · 12×12, 18×18, and 24×24 maps, validation | accuracy: 87.5 ± 0.2 -> 91.7 ± 0.1; 90.6 ± 0.5 -> 94.5 ± 0.1; 92.8 ± 0.2 -> 96.3 ± 0.2 | Helped |
| Train with supervised learning that ignores the bilevel problem structure | Shortest Path and TSP with Interdiction · Validation | accuracy: 26.2 ± 2.4, 20.2 ± 0.5, 14.0 ± 1.0; 10.7 ± 3.9, 9.4 ± 0.4, 9.3 ± 1.2 | Hurt |
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